The Master Formula Box
Mean ($\bar{x}$): $\frac{\Sigma x_i}{n}$Variance ($\sigma^2$): $\frac{\Sigma d_i^2}{n} - (\frac{\Sigma d_i}{n})^2$ (where $d_i = x_i - A$)Coefficient of Variation: $\frac{\sigma}{\bar{x}} \times 100$
Dimensional Analysis of Statistics
The Mean, Median, and Standard Deviation have the same units as the original data $[X]$. The Variance has units of $[X^2]$. If you are measuring the height of students in meters ($m$), the SD will be in $m$, but the Variance will be in $m^2$. Always check this to ensure you haven't forgotten to take the square root!
Variations: Change of Origin and Scale
If $y_i = \frac{x_i - A}{h}$:
- New Mean: $\bar{y} = \frac{\bar{x} - A}{h}$
- New Variance: $\sigma_y^2 = \frac{\sigma_x^2}{h^2}$
- New SD: $\sigma_y = \frac{\sigma_x}{|h|}$
Shortcuts & Mnemonics
- Root Mean Square (RMS): In Physics (AC Circuits), the $V_{rms}$ is essentially the Standard Deviation of the voltage signal.
- The 68-95-99.7 Rule: In a normal distribution, $68\%$ of data falls within $1\sigma$, $95\%$ within $2\sigma$, and $99.7\%$ within $3\sigma$.
- Mnemonic: "Mean is the middle, Variance is the wiggle."
Edge Cases
- Variance = 0: This only happens if all data points are identical. There is no "wiggle."
- Standard Deviation of First $n$ Natural Numbers: $\sigma = \sqrt{\frac{n^2 - 1}{12}}$. A very popular JEE shortcut!
- Combined Variance: When two groups are merged, the new variance is not the average of the two; it depends on the difference between their means.
